4.5 First-order linear equations  (Page 6/10)

$\frac{dy}{dx}=x^{2}y+\text{sin}x$

$\frac{dy}{dt}=ty$

$\frac{dy}{dt}+y^2=x$

$y\prime =x^3+e^x$

$y\prime =y+e^y$

$y\prime =x^{3}y+\text{sin}x$

$y\prime +3y-\text{ln}x=0$

$\text{−}xy\prime =\left(3x+2\right)y+x{e}^x$

$\frac{dy}{dt}=4y+ty+\text{tan}t$

$\frac{dy}{dt}=yx\left(x+1\right)$

$y\prime =xy+3$

$y\prime +e^{x}y=\text{sin}x$

$y\prime =x\text{ln}\left(x\right)y+3x$

$\frac{dy}{dx}=\text{tanh}\left(x\right)y+1$

$\frac{dy}{dt}+3ty=e^{t}y$

$y\prime =3y+2$

$y\prime =2y-x^2$

$xy\prime =3y-6x^2$

$\left(x+2\right)y\prime =3x+y$

$y\prime =3x+xy$

$xy\prime =x+y$

$\text{sin}(x)y\prime =y+2x$

$y\prime =y+e^x$

$xy\prime =3y+x^2$

$y\prime +\text{ln}x=\frac{y}{x}$

[T] $(x+2)y\prime =2y-1$

[T] $y\prime =3e^{t\text{/}3}-2y$

[T] $xy\prime +\frac{y}{2}=\text{sin}(3t)$

[T] $xy\prime =2\frac{\text{cos}x}{x}-3y$

[T] $(x+1)y\prime =3y+x^2+2x+1$

[T] $\text{sin}(x)y\prime +\text{cos}(x)y=2x$

[T] $\sqrt{x^2+1}y\prime =y+2$

[T] $x^{3}y\prime +2x^{2}y=x+1$

$y\prime +y=x,y(0)=3$

$y\prime =y+2x^2,y(0)=0$

$xy\prime =y-3x^3,y(1)=0$

$x^{2}y\prime =xy-\text{ln}x,y(1)=1$

$\left(1+x^2\right)y\prime =y-1,y(0)=0$

$xy\prime =y+2x\text{ln}x,y(1)=5$

$(2+x)y\prime =y+2+x,y(0)=0$

$y\prime =xy+2x{e}^x,y(0)=2$

$\sqrt{x}y\prime =y+2x,y(0)=1$

$y\prime =2y+x{e}^x,y(0)=\mathrm{-1}$

A falling object of mass $m$ can reach terminal velocity when the drag force is proportional to its velocity, with proportionality constant $k.$ Set up the differential equation and solve for the velocity given an initial velocity of $0.$

Using your expression from the preceding problem, what is the terminal velocity? ( Hint: Examine the limiting behavior; does the velocity approach a value?)

[T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall $5000$ meters if the mass is $100$ kilograms, the acceleration due to gravity is $9.8$ m/s ² and the proportionality constant is $4?$

A more accurate way to describe terminal velocity is that the drag force is proportional to the square of velocity, with a proportionality constant $k.$ Set up the differential equation and solve for the velocity.

Using your expression from the preceding problem, what is the terminal velocity? ( Hint: Examine the limiting behavior: Does the velocity approach a value?)